Depletion Zone

Part of The Transistor

The carrier-free region at p-n junctions — its formation, width, and role in transistor operation.

Why This Matters

The depletion zone is the physical mechanism that makes p-n junction devices work. It is the barrier at the heart of every diode, the region that must be overcome for a transistor to conduct, and the source of the junction capacitance that limits high-frequency performance. Without understanding the depletion zone, device behavior is a collection of empirical rules; with it, every major device characteristic can be derived from first principles.

For transistor operation specifically, the two depletion zones at the emitter-base and collector-base junctions have opposite functions: the emitter-base depletion zone is reduced under forward bias (enabling carrier injection), while the collector-base depletion zone is widened under reverse bias (sweeping injected carriers into the collector circuit). The interplay of these two zones is transistor amplification.

For a rebuilding civilization fabricating transistors, understanding the depletion zone helps diagnose problems: too-wide depletion zone at the collector junction means high capacitance and poor high-frequency response. Punch-through (depletion zone at collector extending all the way through the base to the emitter) means the transistor cannot be turned off. Both have specific causes and specific fixes.

Formation at Equilibrium

When p-type and n-type semiconductor are joined, mobile carriers diffuse across the boundary:

• Electrons from n-side move to p-side
• Holes from p-side move to n-side

Diffusing electrons leave behind positively charged donor ions in the n-side. Diffusing holes leave behind negatively charged acceptor ions in the p-side. These fixed charges create an electric field from n to p.

The field opposes further diffusion. Equilibrium is reached when diffusion tendency equals field repulsion. The region containing these fixed charges — depleted of mobile carriers — is the depletion zone.

Width in the two regions: The depletion zone extends further into the more lightly doped side. If p-side doping NA = 10^17 and n-side doping ND = 10^15, most of the depletion zone is on the n-side (lower doping = more width needed to provide the same charge).

Width relationships:

• xp = W × ND / (NA + ND) (depletion width into p-side)
• xn = W × NA / (NA + ND) (depletion width into n-side)
• Total: W = xp + xn = √(2ε × V_total × (NA + ND) / (q × NA × ND))

where V_total = V_bi - V_applied (built-in potential minus applied voltage).

For silicon junction with NA = ND = 10^16 cm^-3 and V_bi = 0.7V: W = √(2 × 11.7 × 8.85×10^-14 × 0.7 × 2×10^16 / (1.6×10^-19 × 10^32)) ≈ 0.3 µm

For lightly doped germanium (ND = 10^14 cm^-3): W ≈ several micrometers

Depletion Zone Under Bias

Forward bias (reduce barrier): V_applied > 0, V_total = V_bi - V_applied decreases. W decreases. At V_applied = V_bi (0.7V for silicon), W → 0 theoretically. In practice, current becomes very large before V_bi is fully overcome.

The shrinking depletion zone means the barrier against carrier injection is reduced, allowing exponential current increase with applied voltage.

Reverse bias (increase barrier): V_applied < 0, V_total = V_bi + |V_reverse| increases. W increases. For large reverse bias (10-100V), V_total >> V_bi and W scales approximately as √V_reverse.

The growing depletion zone creates a wider barrier. No majority carriers can cross; only thermally generated minority carriers that happen to be swept across by the field contribute to (small) leakage current.

Punch-Through in Transistors

In a BJT, the base is sandwiched between emitter and collector. Both junctions have depletion zones that extend into the base:

• Emitter-base depletion zone extends xn_EB into the base (n-side of EB junction)
• Collector-base depletion zone extends xp_CB into the base (p-side of CB junction for PNP)

The actual conducting base width = physical base width WB - xn_EB - xp_CB.

As collector reverse voltage increases, the collector-base depletion zone grows, eating into the base. At high enough voltage, the collector depletion zone reaches the emitter depletion zone: punch-through. When punch-through occurs, carriers can pass directly from emitter to collector through the merged depletion zones — the transistor conducts uncontrollably regardless of base current.

Punch-through voltage BVPT < BVCEO. Lightly-doped narrow bases have lower punch-through voltage. For thin-base high-gain transistors, punch-through is the limiting factor, not avalanche breakdown.

Prevention: Keep base doping sufficiently high so the collector depletion zone occupies only the collector side and a small fraction of the base. Heavier base doping reduces depletion zone width on the base side (xp_CB becomes small).

Design rule: ensure that at maximum reverse collector voltage, the sum of depletion zone extensions into the base does not exceed 70% of WB.

Junction Capacitance

The depletion zone acts as a capacitor: parallel plates of fixed charge separated by the insulating depletion region.

C_j = ε × A / W = C_j0 / √(1 + V_R / V_bi)

where A is junction area, W is depletion width (voltage dependent), and C_j0 is capacitance at zero bias.

For a silicon junction of area 1 mm² and W = 0.3 µm: C_j = 11.7 × 8.85×10^-14 × 10^-2 / 3×10^-5 ≈ 35 pF.

Role in transistor speed: In the common-emitter configuration, the collector-base capacitance C_µ = C_j(CB) is the Miller capacitance. The Miller effect multiplies it by (1 + |Av|). For Av = -100 and C_µ = 5 pF: Miller capacitance at input = 505 pF. This limits high-frequency gain.

Reduce C_µ by: smaller junction area (reduces collector current capacity), higher reverse bias (widens depletion zone, reduces C_µ), or operating at lower gain (reduces Miller multiplication).

Varactor diodes: A reverse-biased p-n junction is a voltage-controlled capacitor (varactor). Deliberately exploiting C_j(V) dependence enables electronically tuned resonant circuits. Varactor diodes replace mechanical capacitor adjustment in tuned circuits for AM/FM radio receivers and transmitters. Tuning voltage controls the resonant frequency: f = 1/(2π√(LC)) where C = C_j(V_tune).

Depletion Zone in Transistor Switching

When a transistor switches from saturation (fully on) to cutoff (off), both depletion zones must establish their equilibrium reverse-biased widths. This requires sweeping minority carriers out of the base and quasi-neutral regions — “charge storage.”

The storage charge in the base during saturation: QB ≈ τ_B × IC_sat, where τ_B is base minority carrier lifetime. For τ_B = 1 µs and IC_sat = 10 mA: QB = 10 nC. To remove this charge, a reverse base current flows for a time: t_storage ≈ QB / I_B_reverse.

Speed-up (anti-saturation): Apply a Schottky clamp — a Schottky diode between base and collector of the transistor. When the transistor would saturate, the Schottky diode conducts (its forward voltage ~0.3V is less than VBE = 0.7V), diverting base current and preventing full saturation. The transistor stays at the edge of saturation rather than deep in it. Storage charge is minimal; switching speed increases from microseconds to nanoseconds.

This anti-saturation technique is the reason TTL logic (Transistor-Transistor Logic) gates with Schottky diodes (74S and 74LS series) were so much faster than earlier TTL without Schottky clamps. For a rebuilding civilization building discrete logic circuits, the Schottky-clamped transistor switch is the path to fast digital logic.